hlab meeting -- paper -- t.gogami 30apr2013. experiments with magnets (e,ek + ) reaction
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HLAB MEETING-- Paper --
T.Gogami30Apr2013
Experiments with magnets(e,e’K+) reaction
• Dispersive plane• Transfer matrix• R12 , R16
• Emittance• Beam envelope• ・・・
詳細な計算 [参照 ]Transport AppendixK.L.Brown and F.Rothacker
Paper
Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
Design requirements
1. Correct beam transport properties2. To reduce the – Weight– Cost– Power
Dipole, Quadrupole, Sextupole
By(x) = a + bx + cx2 + ・・・・The field of the magnet as a multpole expansion about the central trajectory
Dipole term Quadrupole term Sextupole term
Dipole elements
R0 = mv/qB0
ObjectImage
Particle of higher momentum
Dipole termQuadrupole termSextupole term
Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
Field-path integral
Field-path integral B0R0
1 rad
𝑅0=𝑝
𝐵0𝑞
[rad]
Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
A quadropole element
A) By a separate quadrupole magnetB) By a rotated input or output in a bending magnetC) By a transverse field gradient in a bending magnet
A quadropole element
A) By a separate quadrupole magnetB) By a rotated input or output in a bending magnetC) By a transverse field gradient in a bending magnet
Extra cost
Rotated pole edge (1)
Imaging in the dispersive plane( Frequently used to generate first order imaging )
ΔB y=𝜕𝐵 𝑦𝜕 𝑥
𝑥
−𝜃=−𝑠 Δ𝐵 𝑦𝐵0𝑅 0
=−x
𝜕𝜕𝑥
(𝐵 𝑦 𝑠)
𝐵0𝑅 0
B y s=−𝐵0𝑥 tan𝛼
1𝑓 𝑥
=−𝜃𝑥
=−tan𝛼𝑅0
−𝜃=−𝑥tan𝛼𝑅0
Optical focusing power
Rotated pole edge (2)( Frequently used to generate first order imaging )
Imaging in the non-dispersive plane
ΔB x=𝜕𝐵𝑥𝜕 𝑦
𝑦=𝜕𝐵 𝑦𝜕𝑥
𝑦
𝜑=𝑠 Δ𝐵𝑥𝐵0𝑅 0
=𝑦
𝜕𝜕 𝑥
(𝐵 𝑦 𝑠)
𝐵0𝑅 0
B y s=−𝐵0𝑥 tan𝛼
1𝑓 𝑦
=−𝜑𝑦
=tan𝛼𝑅0
𝜑=−𝑦tan𝛼𝑅0
(Rot B = 0 )
Rotated pole edge (3)( Frequently used to generate first order imaging )
1𝑓 𝑥
=−tan𝛼𝑅0
Optical focusing power
Dispersive plane
Non-dispersive plane
Transverse field gradient (1)
𝑑𝑥 ′𝑥
=−𝑑𝑠𝑅02
1𝑓 𝑥
=𝑑𝑠𝑅02
Focusing power
𝑑𝑥 ′𝑥
=−𝜕𝐵 𝑦𝜕 𝑥
𝑑𝑠𝐵0𝑅 0
=𝑛
𝑅02𝑑𝑠
Transverse field gradient is zero (Pure dipole field)
Transverse field gradient is not zero
dB y=𝜕𝐵 𝑦𝜕 𝑥
𝑥
𝑛=−𝑅0𝐵0
𝜕𝐵 𝑦𝜕 𝑥 Field index
Transverse field gradient (2)
Total focusing power ( Dipole + transverse field gradient )
𝑛=−𝑅0𝐵0
𝜕𝐵 𝑦𝜕 𝑥
Field index
A) A pure dipole filedFocusing in the dispersive plane
B) A transverse field gradient characterized by n– Focusing in both plane– Sum of the focusing powers is constant
1/fx + 1/fy = (1-n)/(R02)ds – n/R0
2 = ds/R02
C) If n=1/2Dispersive and non-dispersive focusing power: ds/2R0
2
D) If n < 0– Dispersive plane focusing power : strong and positive– Non-dispersive plane focusing power : negative
Transverse field gradient (3)
𝑛=−
𝑅0𝐵0
𝜕𝐵 𝑦𝜕 𝑥
Field index
Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
Matrix formalism (first order)
x1 = x
x2 = θ = px/pz(CT)
x3 = y
x4 = φ = py/pz(CT)
x5 = l = z – z(CT)
x6 = δ = (pz – pz(CT))/pz(CT)
𝑥𝑖 (𝑠 )=∑𝑗=1
6
𝑅𝑖𝑗 𝑥 𝑗 (0)
Examples of transport matrices Rij
Imaging
• R12 = 0– x-image at s with magnification R11
• R34 = 0– y-image at s with magnification R33
Focal lengths and focal planes
• x-plane
• y-plane
Dispersion
Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
Phase ellipse and Beam envelope
√𝜎 22
√𝜎 11x
θ
Phase ellipse
Beam emittance
x
z
√𝜎 11
1/2
Beam Envelope
s = 0 beam size (beam waist)
Output beam matrix
• Initial beam ellipse• R-matrix
σ (0 )=(𝜎 11(0) 00 𝜎 22(0))
𝜎 (𝑠 )=𝑹 σ (0 ) 𝑹𝑇=(𝜎 11(𝑠) 𝜎 12(𝑠)𝜎12(𝑠) 𝜎 22(𝑠))
𝜎 22 (𝑠) 𝑥2−2𝜎 12 (2 )𝑥𝜗+𝜎 11 (𝑠 )𝜗 2=|𝜎||𝜎|=𝜎 11 (𝑠)𝜎 22 (𝑠)−𝜎 212(𝑠)=𝜎 11(0)𝜎 22(0)
Initial Beam matrix
After a magnet system with an R-matrix (Rij)
Output beam ellipse
• Final beam matrix• Final beam ellipse
Contents
• Introduction• Field-path integrals• First order imaging• Matrix formalism• Beam envelope and phase ellipse• Second order aberrations and sextupole elements• Practical magnet design
Parameters
Practical magnet design
A) Bending power
B) Pole gap
C) Coil power
D) Magnet weight : Coil weight
: Steel weight
Key constrains
An advantage B0
R0 Focal length
“Strong focusing” techniqueLarge pole edge rotation + Large field index
NOVA NV-10 ion implanter
Bend : 70 degreesGap : 5 cmBending radius : 53.8 cmPole gap field : 8 kGParticle : 80 keV antimonyWeight : 2000 lbPole edge rotation : 35 degreesField index : -1.152
x-defocusy-focus
x-focusy-defocus
x : DFDy : FDF
Uniform field bending magnet• Weight : 4000 lb• Pole gap field : 16 kG• Coil power : substantially higher
SPL with field clamp + ENGE
New magnetic field map Committed to the svn
Split pole magnet (ENGE)
Matrix tuning (E05-115)
Before
After
FWHM ~ 4 MeV/c2
Backup
Transverse field gradient (2)
Total focusing power ( Dipole + transverse field gradient )
𝑛=−𝑅0𝐵0
𝜕𝐵 𝑦𝜕 𝑥
Field index
Simple harmonic motion
Simple harmonic motion
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